A007615 Primes with unique period length (the periods are given in A007498). (Formerly M2890)

3, 11, 37, 101, 333667, 9091, 9901, 909091, 1111111111111111111, 11111111111111111111111, 99990001, 999999000001, 909090909090909091, 900900900900990990990991, 9999999900000001, 909090909090909090909090909091, 900900900900900900900900900900990990990990990990990990990991

Offset

1,1

Comments

Additional terms are Phi(n,10)/gcd(n,Phi(n,10)) for the n in A007498, where Phi(n,10) is the n-th cyclotomic polynomial evaluated at 10.

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Samuel Yates, Period Lengths of Exactly One or Two Prime Numbers, J. Rec. Math., 18 (1985), 22-24.

Links

Max Alekseyev, Table of n, a(n) for n = 1..98 (terms 1..25 from T. D. Noe; terms 26..31 from Ray Chandler)

C. K. Caldwell, The Prime Glossary, unique prime

Makoto Kamada, Factorizations of Phi_n(10)

Index entries for sequences related to decimal expansion of 1/n

Formula

a(n) = A061075(A007498(n)). - Max Alekseyev, Oct 16 2010

a(n) = A006530(A019328(A007498(n))). - Ray Chandler, May 10 2017

Example

3 is the only prime p such that decimal expansion of 1/p has (nontrivial) period exactly 1.

Mathematica

nmax = 50; periods = Reap[ Do[ p = Cyclotomic[n, 10] / GCD[n, Cyclotomic[n, 10]]; If[ PrimeQ[p], Sow[n]], {n, 1, nmax}]][[2, 1]]; Cyclotomic[#, 10] / GCD[#, Cyclotomic[#, 10]]& /@ periods // Prepend[#, 3]& (* Jean-François Alcover, Mar 28 2013 *)

Crossrefs

Cf. A007498, A040017, A002371, A048595, A006883, A007732, A051626, A061075, A006530, A019328.

Sequence in context: A061075 A005422 A040017 * A065540 A084171 A192875

Adjacent sequences: A007612 A007613 A007614 * A007616 A007617 A007618

Keyword

nonn,nice,easy,base

Author

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Status

approved